Combinatorics, Complexity and Randomness (Turing Award by Karp R.

By Karp R.

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Chapter 2 One Step at a Time. 1 Weak Induction Let us assume it is almost midnight, and it has not rained all day today. If, from the fact that it does not rain on a given day, it followed that it will not rain the following day, it would then also follow that it would never rain again. Indeed, from the fact that it does not rain today, it would follow that it will not rain tomorrow, from which it would follow that it will not rain the day after tomorrow, and so on. This simple logic leads to another very powerful tool in mathematics: the method of mathematical induction.

If we temporarily disregard A, we have n players left, so by the induction hypothesis there will be one of them, say B, who will list the names of the other n— 1 players. Now if B defeated A, or if anyone defeated by B defeated A, then B lists the name of A, too, and we are done. If not, then A has defeated B, and all the players defeated by B, so A won more games than B, a contradiction. (3) Induction on n. For n = 1, the statement is trivially true. Now assume the statement is true for n and prove it for n + 1.

In other words, a bijection matches the elements of X with the elements of Y, so that each element will have exactly one match. 9. Let / : X —> Y be a function. 8, then we say that / is one-to-one or injective, or is an injection. 8, then we say that / is onto or surjective, or is a surjection. 10. Let X and Y be two finite sets. If there exists a bijection f from X onto Y, then X and Y have the same number of elements. Proof. The bijection / matches elements of X to elements of Y, in other words it creates pairs with one element from X and one from Y in each pair.

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